The size of the optimal order quantity depends on. Determining the optimal delivery lot size

Let us consider the operation of a warehouse in which inventories are stored and used to supply consumers. The operation of a real warehouse is accompanied by many deviations from the ideal regime: a batch of one volume was ordered, but a batch with a different volume arrived; According to the plan, the shipment should arrive in two weeks, but it arrived in 10 days; with the unloading rate of one day, the unloading of the batch lasted three days, etc. It is almost impossible to take into account all these deviations, therefore, when modeling warehouse operation, the following assumptions are usually made.

• 1. The rate of consumption of stocks from the warehouse is a constant value, which we denote M(units of inventory per unit of time); in accordance with this, the graph of changes in the amount of reserves in terms of consumption is a straight line segment.
• 2. Volume of replenishment batch Q there is a constant value, so the inventory management system is a fixed order quantity system.
• 3. The unloading time of the arriving replenishment batch is short; we will consider it equal to zero.
• 4. The time from making a decision on replenishment to the arrival of the ordered batch is a constant value Δ t, so we can assume that the ordered batch arrives instantly: if you need it to arrive exactly at a certain moment, then it should be ordered at the time of At previously.
• 5. There is no systematic accumulation or overexpenditure of inventory in the warehouse. If through T to indicate the time between two consecutive deliveries, then the equality must be fulfilled: Q = MT. From the above it follows that the warehouse operates in identical cycles lasting T, and during the cycle the stock value changes from the maximum level S to the minimum level s.
• 6. Finally, we will consider it mandatory to fulfill the requirement that the lack of inventory in the warehouse is unacceptable, i.e. inequality holds s> 0. From the point of view of reducing warehouse storage costs, it follows that s= 0 and therefore S = Q.

The final schedule of ideal warehouse operation in the form of a dependence of the amount of inventory at from time t will have the form shown in Fig. 12.3.

It was previously noted that the efficiency of a warehouse is assessed by its costs of replenishment and storage. Costs that do not depend on the volume of the batch are called invoices. This includes postal and telegraph costs, travel expenses, some transport costs, etc. We will denote overhead costs by TO. We will consider the costs of storing inventories to be proportional to the size of the stored inventories and the time of their storage. The cost of storing one unit of inventory for one unit of time is called the value of specific storage costs; we will denote them by h.

Rice. 12.3.

When the amount of stored inventory changes, storage costs over time T is obtained by multiplying the quantity h And T by the average value of inventories during this time T. Thus, the warehouse costs over time T with replenishment batch size Q in the case of an ideal warehouse operating mode shown in Fig. 12.3 are equal

After dividing this function by a constant value T subject to equality Q = MT we obtain an expression for the cost of replenishment and storage of inventories per unit of time:

This will be the objective function, the minimization of which will allow us to indicate the optimal operating mode of the warehouse.

Let's find the volume of the ordered batch Q, at which the function of average warehouse costs per unit of time is minimized, i.e. function . On practice Q often take discrete values, in particular due to the use of vehicles of a certain carrying capacity; in this case the optimal value Q found by searching through acceptable values Q. We will assume that the restrictions on the accepted values Q no, then the problem for the minimum of a function (it is easy to show that it is convex, Fig. 12.4 can be solved using differential calculus methods:

where we find the minimum point:

This formula is called Wilson's formula(named after the English eco-scientist who received it in the 20s of the last century).

The optimal batch size, calculated using Wilson's formula, has the characteristic property: batch size Q is optimal if and only if the cycle-time storage costs T equal to overhead TO.

Rice. 12.4.

Indeed, if,then storage costs

per cycle are equal

If storage costs per cycle are equal to overhead costs, i.e.

Let us illustrate the characteristic property of the optimal batch size graphically.

In Fig. 12.4 it is clear that the minimum value of the function is achieved at that value Q, at which the values ​​of the other two functions that make up it are equal.

Using Wilson's formula (12.18), under the previously made assumptions about ideal warehouse operation, a number of calculated characteristics of warehouse operation in optimal mode can be obtained.

Optimal average stock level:

Optimal replenishment frequency:

Optimal average costs of storing inventory per unit time:

(12.21)

Example

Let's consider a typical problem. Cement is delivered to the warehouse by barge in quantities of 1,500 tons. Consumers pick up 50 tons of cement from the warehouse per day. Overhead costs for delivering a batch of cement are 2 thousand rubles. The cost of storing 1 ton of cement during the day is equal to 0.1 rubles. It is required to determine: 1) cycle time, average daily overhead costs and average daily storage costs; 2) the same values ​​for batch sizes of 500 tons and 3000 tons; 3) the optimal size of the ordered batch and the calculated characteristics of the warehouse in optimal mode.

Warehouse operating parameters:

1. Cycle duration ( T):

Average daily overhead costs:

Average daily storage costs:

2. We will carry out similar calculations for t:

3. Let’s find the optimal size of the ordered batch using Wilson’s formula (12.18):

The optimal average stock level is calculated using formula (12.19):

The optimal frequency of inventory replenishment is calculated using formula (12.20):

To calculate the optimal average costs of storing inventory per unit of time, formula (12.21) is used.

Inventories play both a positive and negative role in the activities of the logistics system. The positive role is that they ensure the continuity of the processes of production and sales of products, being a kind of buffer that smoothes out unforeseen fluctuations in demand, delays in the delivery of resources, and increase the reliability of logistics management.

The negative side of creating reserves is that they immobilize significant financial resources that could be used by the enterprise for other purposes, for example, investing in new technologies, market research, improving the economic performance of the enterprise.

In addition, large levels of inventory of finished products impede the improvement of their quality, since the enterprise is primarily interested in selling existing products before investing in improving their quality. Based on this, the problem arises of ensuring the continuity of logistics and technological processes at a minimum level of costs associated with the formation and management of various types of inventories in the logistics system.

One of the methods of effective inventory management is to determine the optimal shipments of cargo, which allows you to optimize the costs of transportation, storage of cargo, and also to avoid excess or shortage of cargo in the warehouse.

Optimal batch size q determined by the criterion of minimum costs for transporting products and storing inventories.

The amount of total costs is calculated using formula (3.1):

where n is the number of shipments delivered during the billing period

where is the average amount of stock (in tons), which is determined on the assumption that a new batch is imported after the previous one is completely consumed.

In this case, the value is calculated using the following formula:

The total cost function C has a minimum at the point where its first derivative with respect to q is equal to zero, i.e.

As the size of the annual volume of product consumption, we accept the data obtained as a result of forecasting using the regression analysis method: thousand tons/year; tariff for transportation of one batch of c.u./t; costs associated with storing inventory cu/t.

Substituting the given values, we get:

The total costs will be:

Solution to this problem graphically consists in constructing graphs of the dependence, and, having previously performed the necessary calculations to determine, and.

Let's determine the value, and when changing q in the range from 900 to 800 in increments of 1200. The result of the calculations will be entered into Table 3.1.

Table 3.1.

Values ​​, and

Based on the data in Table 3.1, graphs of the dependence of costs (transport, storage and total) on the batch size were constructed (Fig. 3.1).

Figure 3.1 Dependence of costs on batch size

Analysis of the graphs in Figure 3.1 shows that transportation costs decrease with increasing batch size, which is associated with a decrease in the number of flights. Storage costs increase in direct proportion to batch size.

The graph of total costs has a minimum at the value q approximately equal to 993 tons, which is the optimal value for the size of the delivery lot. The corresponding minimum total costs are 9937 USD.

Let's calculate the optimal batch size in conditions of shortage with the amount of expenses associated with the shortage

In conditions of shortage, the value calculated using formula (3.8) is adjusted by the coefficient k, taking into account the costs associated with the deficit.

The amount of spending associated with the deficit;

we accept

Substituting the values, we get:

It follows from this that in conditions of a possible shortage, the size of the optimal delivery lot must be increased by 15%.

Determining the optimal batch size
Dmitry Ezepov, purchasing manager at Midwest © LOGISTIC&system www.logistpro.ru

One of the most difficult tasks for any purchasing manager is choosing the optimal order size. However, there are very few real tools to facilitate its solution. Of course, there is the Wilson formula, which is presented in theoretical literature as such a tool, but in practice its use must be adjusted

The author of this article, working in several large trading companies in Minsk, never saw Wilson’s formula applied in practice. Its absence in the arsenal of purchasing managers cannot be explained by their lack of analytical skills and abilities, since modern companies pay great attention to the qualifications of their employees.

Let's try to find out why “the most common tool in inventory management” does not go beyond scientific publications and textbooks. Below is the well-known Wilson formula, using which it is recommended to calculate the economic order quantity:

where Q is the volume of the purchase batch;

S – the need for materials or finished products for the reporting period;

O – fixed costs associated with fulfilling one order;

C – costs of storing a unit of inventory for the reporting period.

The essence of this formula comes down to calculating what batch sizes should be (all the same) in order to deliver a given volume of goods (that is, the total demand for the reporting period) during a given period. In this case, the sum of fixed and variable costs should be minimal.

The problem being solved has at least four initial conditions: 1) a given volume that needs to be delivered to its destination; 2) specified period; 3) equal batch sizes; 4) pre-approved composition of fixed and variable costs. This formulation of the problem has little in common with the real conditions of doing business. No one knows the capacity and dynamics of the market in advance, so the sizes of ordered batches will always be different. There is also no point in setting a period for planning purchases, since commercial companies usually exist much longer than the reporting period. The composition of costs is also subject to change due to the influence of many factors.

In other words, the conditions for applying the Wilson formula simply do not exist in reality, or at least occur very rarely. Do commercial companies need to solve a problem with such initial conditions? I think not. That is why the “common tool” is implemented only on paper.

WE CHANGE THE CONDITIONS

In market conditions, sales activity is inconsistent, which inevitably affects the supply process. Therefore, both the frequency and size of purchased lots never coincide with their planned indicators at the beginning of the reporting period. If you focus solely on the plan or long-term forecast (as in Wilson’s formula), then one of two situations will inevitably arise: either an overflow of the warehouse or a shortage of products. The result of both will always be a decrease in net profit. In the first case, due to an increase in storage costs, in the second, due to a shortage. Therefore, the formula for calculating the optimal order size must be flexible in relation to the market situation, that is, based on the most accurate short-term sales forecast.

The total costs of purchasing and storing inventories consist of the sum of these same costs for each purchased batch. Consequently, minimizing the cost of delivery and storage of each batch separately leads to minimization of the supply process as a whole. And since calculating the volume of each batch requires a short-term sales forecast (and not for the entire reporting period), the necessary condition for the flexibility of the formula for calculating the optimal batch size (OPS) in relation to the market situation is met. This condition of the problem corresponds to both the goal of a commercial company (minimizing costs) and the real conditions of doing business (variability of market conditions). Definitions of fixed and variable costs for the supply minimization approach on a lot-by-lot basis are provided in the “Types of Costs” box on page 28.

ACTUAL CALCULATION

If we assume that the loan is repaid as the cost of inventory decreases at planned intervals (days, weeks, month, etc.) (1), then, using the formula for the sum of the terms of an arithmetic progression, we can calculate the total cost of storing one batch of inventory (usage fee credit):

where K is the cost of storing inventory;

Q – purchase batch volume;

p – purchase price of a unit of goods;

t is the time the stock is in the warehouse, which depends on the short-term forecast of sales intensity;

r – interest rate per planned unit of time (day, week, etc.).

Thus, the total costs for delivery and storage of the order batch will be:

where Z is the total cost of delivery and storage of the batch.

There is no point in minimizing the absolute value of the cost of delivery and storage of one batch, since it would be cheaper to simply refuse purchases, so you should move on to the relative cost per unit of inventory:

where z is the cost of replenishment and storage of a unit of stock.

If purchases are made frequently, then the sales period for one batch is short, and the sales intensity during this time will be relatively constant2. Based on this, the time the stock is in the warehouse is calculated as:

where is a short-term forecast of average sales for a planned unit of time (day, week, month, etc.).

The designation is not accidental, since the forecast is usually average sales in the past, taking into account various adjustments (shortages in stock in the past, the presence of a trend, etc.).

Thus, substituting formula (5) into formula (4), we obtain the objective function for minimizing the cost of delivery and storage of a unit of inventory:

Equating the first derivative to zero:

we find (ORP) taking into account short-term sales forecast:

NEW WILSON FORMULA

Formally, from a mathematical point of view, formula (8) is the same Wilson formula (the numerator and denominator are divided by the same value depending on the adopted planned unit of time). And if the intensity of sales does not change, say, during the year, then by replacing the annual demand for goods and r with the annual interest rate, we will get a result that will be identical to the calculation of the EOP. However, from a functional point of view, formula (8) demonstrates a completely different approach to the problem being solved. It takes into account the current sales forecast, which makes the calculation flexible relative to the market situation. The remaining parameters of the ORP formula, if necessary, can be quickly adjusted, which is also an undeniable advantage over the classical formula for calculating EOP.

The company's purchasing policy is also influenced by other, often more significant factors than the intensity of sales (current balances in the company's own warehouse, minimum batch size, delivery conditions, etc.). Therefore, despite the fact that the proposed formula eliminates the main obstacle to calculating the optimal order size, its use can only be an auxiliary tool for effective inventory management.

A highly professional purchasing manager relies on a whole system of statistical indicators, in which the ORP formula plays a significant, but far from decisive role. However, the description of such a system of indicators for effective inventory management is a separate topic, which we will cover in the next issues of the magazine

1- In reality this does not happen, so the cost of holding inventory will be higher. 2- In reality, you need to pay attention not to order frequency, but to the stability of sales within the short-term sales forecast period. It’s just that usually, the shorter the period, the less seasonality and tendency appear.

After the choice of a replenishment system has been made, it is necessary to quantify the size of the ordered batch, as well as the time interval after which the order is repeated.

The optimal size of the batch of goods supplied and, accordingly, the optimal frequency of delivery depend on the following factors: volume of demand, costs of delivering goods, costs of storing inventory.

The minimum total costs for delivery and storage are chosen as an optimality criterion.

Both delivery costs and storage costs depend on the size of the order, however, the nature of the dependence of each of these cost items on the order volume is different. The costs of delivering goods as the order size increases obviously decrease, since transportation is carried out in larger quantities and, therefore, less frequently. The graph of this dependence, which has the shape of a hyperbola, is presented in Fig. 60.

Storage costs increase in direct proportion to the size of the order. This dependence is graphically presented in Fig. 61.

Rice. 60. Dependence of transportation costs on order size

Rice. 61. Dependence of inventory storage costs on order size

Rice. 62. Dependence of the total costs of storage and transportation on the size of the order.

By adding both graphs, we obtain a curve reflecting the nature of the dependence of the total costs of transportation and storage on the size of the ordered batch (Fig. 62). As you can see, the total cost curve has a minimum point at which the total costs will be minimal.

The problem of determining the optimal order size, along with the graphical method, can also be solved analytically. Wilson's formula is used for this.

This article does not pretend to give a comprehensive answer to the question of optimal production batch sizes; its purpose is to collect in one place some aspects of one of the problems of planning complex production.

Let's start with the definition

In general, to really start the answer correctly, you need to define a production batch. And this attempt alone can give rise to several crusades and holy wars between adherents of one or another approach. At least in those years when I worked as a consultant in a consulting company, we spent a long time debating this definition, until one of our wise colleagues proposed 5 options that would more or less cover all the many variations of production batches.

The party is:

1. Customer order size – external or internal (between operations)
2. Technological batch – simultaneously processed quantity of products
3. Quantity of products produced between changeovers
4. Quantity of products produced between shipments
5. The volume of the drive or hopper loaded at a time before the operation

In general, we should say that a production batch is the number of parts, products, products that are processed at one stage of production without interruptions, stops and switching to another type of parts, products, products. I can't say that this is the best definition of a party that can be given, but for the purposes of this article, I think it will suffice.

Economically optimal batch size per operation

For each individual stage of production, the economically optimal batch size can be determined quite reliably, for which Wilson’s formula is used

where EOQ is the economic order quantity (EOQ)),
Q - quantity of goods per year (Quantity in annual units),
P - costs of order implementation (Placing an order cost),
C - costs of storing a unit of goods per year (Carry costs)

or its analogue Andler's formula

where y min is the optimal batch size,
V - the required volume of products over a period of time (sales speed),
Cr- costs associated with changing batches (conditionally - for setup),
Cl- specific warehousing costs over a period of time.

The general appearance of the graph is as follows:

Actually, here we need to look for the minimum of the “Total costs” curve, and the value of X that corresponds to it will represent the “economically optimal batch size”.

Naturally, this all looks simple, only on the graph; in order to calculate the exact value, you need to have a good understanding of setup costs (green curve) and the amount of warehouse costs (purple curve).

Setup costs may include:

• cost of equipment downtime
• cost of operator downtime
• costs for adjusters
• tool costs
• tooling costs
• additional costs of materials and energy during shutdown/startup
• etc.

Warehouse costs include:

• cost of stored objects
• cost of warehouse space
• warehouse staff costs
• lighting and heating costs
• costs for warehouse equipment (stackers/loaders)
• etc.

In general, there is quite a lot to consider.

The total cost curve does not have a kink at the current minimum, which means that if you have, for example, an economically optimal batch size of 1327 pieces, then most likely you can run production in batches of 1300 to 1400 pieces without any significant deviations in cost, and certainly if the optimal batch size is 4.6 pieces, then you can launch batches of 4 pieces and 5 pieces.

Problem: different technologies - different batches

The problem with actual production is that set-up and inventory costs are not the same throughout the production cycle, and this introduces variability in the size of batches that go through multiple stages of production rather than just one.

For example, it is profitable to bring raw materials by truck, because... the cost of the vehicle is “spread” over the entire volume of raw materials, no matter how much there is, heat treatment must be performed for as many parts as can be put into the furnace as much as possible, and shipment must be made only in the quantity ordered by a specific customer, otherwise everything is unnecessary, whatever you send him will simply be given to him for free.

Storing small and bulk objects also costs different amounts of money, and if some raw materials also need to be kept warm or in other “special climatic conditions,” then the cost of storing such raw materials will be higher than for other types of raw materials.

1. 2000 pieces per lot
2. 200 pieces per lot
3. 540 pieces per lot
4. 34 pieces per lot

And it’s also good if the units of measurement are the same in each case. Otherwise it could turn out like this:

1. 2000 kg per lot
2. 200 pieces per lot
3. 540 pairs per lot
4. 34 sets per lot

In this case, the problem of optimal batch size only gets worse.

Extreme solutions to the problem

To avoid confusion, you want to have one batch size for all occasions. After all, if at one stage of production a batch consists of ten pieces, and at another of thirteen, it is necessary to organize some kind of intermediate warehouse in order to accumulate the missing pieces of semi-finished products.

What extreme options could there be?

1. use the maximum of the calculated lot sizes
2. use the minimum calculated lot size

Let's take the example with pieces described above (2000, 200, 530 and 34 pieces) and see how to implement both options on it.

Maximum lot size

The maximum lot size of all four options is 2000 pieces. Having agreed to its use, we come to production planning, in which only batches of 2000 pieces are used:

1. 2000 pieces per lot
2. 2000 pieces per lot
3. 2000 pieces per lot
4. 2000 pieces per lot

What happens with this?

At the first stage, we obtain the optimal batch size - no more, no less. And those who work at this site, and even more so those who manage it, should be absolutely satisfied with this decision.

At the second stage, the batch size is 10 times larger than the optimal one. What does this mean? We spend 10 times less time on retooling this stage of production, but at the same time we fill the intermediate warehouse between stages 2 and 3 with a large volume of inventory, which is ten times greater than what our managers would be comfortable with.

At the third stage, the batch size is almost 4 times larger than the optimal one, and this can also lead to a large amount of inventory.

But here’s where there are definitely a LOT of reserves - this is after the fourth stage. There you can work 34 pieces at a time, which means that the batch size is almost 60 times larger than optimal.

What is good and what is bad about this decision.

A good result is that the equipment will be fully loaded, downtime for changeover will be minimized, and if we can synchronize the changeover of equipment and pass one batch through all stages in order, then we will only need three intermediate warehouses for 2000 pieces of semi-finished products (between the first and second stages, between the second and third stages, between the third and fourth stages) and then the whole process will work like a conveyor. If any of the stages stops, then the limit on the size of the intermediate warehouse of 2000 pieces will quickly force the entire production to stop and overproduction will not occur: subsequent stages will exhaust their stocks of semi-finished products and stop, because the emergency stage will not allow them to be replenished, and the previous stages will fill the intermediate warehouses and also stop, because the emergency stage will not allow them to be released).

The bad result is that you will most likely need a lot of warehouse space to organize three intermediate warehouses: this is how production is most often organized. that until all 2000 semi-finished products appear in the previous warehouse, the next stage of production does not start, which means that you need to have appropriate space for these semi-finished products (in some cases you can work “from wheels”, i.e. start production at the next stage yet before the entire batch of 2000 semi-finished products is completed, but this is not possible for every technology). The worst situation will be with the warehouse of finished products, because... there we will get a catastrophic supply of excess production.

Minimum lot size

The minimum lot size of all four options is 34 pieces. Having agreed to use it, we come to production planning, in which only batches of 34 pieces are used:

1. 34 pieces per lot
2. 34 pieces per lot
3. 34 pieces per lot
4. 34 pieces per lot

What happens with this?

At the first stage, changeovers will be performed 60 times more often than required for the optimal option. That's a lot. If each changeover takes a significant amount of time, this can have a catastrophic effect on the productivity of the entire process - it simply will not have time to produce everything that you want to get from it.

Further readjustment will also be performed non-optimally - 6 times more often than required for the optimal option. Even worse, if, for example, when launching each batch, expensive equipment or materials are used that are consumed once for the entire batch, these costs will increase significantly and will place an exorbitant burden on the cost of production.

The same will happen with the third stage, and only at the fourth stage everything will be as it should be.

In general, the entire production process will be slower and will be held back by the stage with the longest changeover time.

The advantages of this option are that you minimize the need for warehouse space - you only need as much as is required to store 3 types of semi-finished products of 34 pieces each, a little more for 34 units of raw materials and 34 units of finished products. Microscopic figure compared to the previous stage.

Disadvantages - increased losses of equipment during changeovers and reduced productivity of the entire process as a whole due to large losses of time for changeover.

Let's leave everything as it is

Now, having understood what happens in extreme cases, we can figure out how production will operate if we leave the batch sizes such that they are equal to the economically optimal batch size of each stage separately:

1. 2000 pieces per lot
2. 200 pieces per lot
3. 540 pieces per lot
4. 34 pieces per lot

So how will this work?

To launch such production, we will need 2000 units of raw materials before the first stage. Then we will be able to carry out adjustments and launch the optimal batch into production and everything will be fine.

After this, 2000 semi-finished products will go to the intermediate warehouse. Of these, only 200 pieces will be selected during the first run in order to optimally begin the second stage of production. Everything is fine here too.

After the second stage, 200 pieces will go into stock and will wait for the next batch, since at least 540 pieces are needed to launch the third stage. And if the second stage produces semi-finished products of the same type, then two more batches of 200 pieces will need to be produced. In this case, the inventory between the second and third stages will reach 600 pieces and it will be possible to start the third stage of production.

The third stage of production will deliver 540 semi-finished products to the last intermediate warehouse and they will be consumed from there in small batches of 34 pieces. In this case, we will ensure minimal stocks in the warehouse of finished products, but still will not get rid of stocks in the warehouse of semi-finished products between the 3rd and 4th stages of production.

What can you see in this situation?

The size of the intermediate warehouse is proportional to whichever of the economically optimal lots of these two stages is larger in quantity.

Those. The warehouse for semi-finished products between the first and second stages of production must accommodate at least 2000 products. The warehouse for semi-finished products between the second and third stages of production should accommodate 540, and not 200 products. And the warehouse for semi-finished products between the third and fourth stages of production should also accommodate 540 products. The finished products warehouse must accommodate batches of 34 finished products and this, apparently, will be sufficient in our case.

Interestingly, this leads to the first change worth making to the planning system.

Since our warehouse size is larger than optimal (2000, 540, 540 and 34), there is no logical sense in launching batches of 200 pieces at the second stage rather than 540 - we still pay for the warehouse as “for 540” and accumulate parts there to launch at the next stage (at least) 540 pieces, so it is worth changing the size of the economically optimal batch of the second stage from 200 to 540, despite the fact that we obtained the figure 200 by calculation using the above formula.

In reality, making such a decision looks like this: the foreman of the site where the second stage of production is carried out looks at the statistics of stocks of semi-finished products in both warehouses and says something like the following: “why are we even bothering and doing changeovers all the time, no one needs this! »

Thus, we smoothly move on to option 2:

1. 2000 pieces per lot
2. 540 pieces per lot
3. 540 pieces per lot
4. 34 pieces per lot

And this is not arbitrariness, this is simply the common sense of the foreman or planner, because in this case, working in batches of 200 pieces is really not needed for anything other than to comply with the calculated economically justified batch size. And if this is not a game situation, but a real-life situation, then no one gives a damn about the calculated numbers - after all, it is obvious that in this case the calculations did not take into account the features of the entire process.

To illustrate this approach with another example, let's assume that production consists of 10 steps rather than 4, and the optimal batches for each step were calculated as follows:

1. 4000 pieces
2. 70 pieces
3. 320 pieces
4. 15 pieces
5. 645 pieces
6. 90 pieces
7. 425 pieces
8. 64 pieces
9. 130 pieces
10. 70 pieces

Obviously, the reserves between stages should contain no less than:

• 4000 products between the first and second stages
• 320 products between the second and third stages
• 320 products between the third and fourth stages
• 645 products between the fourth and fifth stages
• 645 products between the fifth and sixth stages
• 425 products between the sixth and seventh stages
• 425 products between the seventh and eighth stages
• 130 products between the eighth and ninth stages
• 130 products between the ninth and tenth stages

After thinking a little about the optimal batch sizes, you can come to the conclusion that you can just as well set the batch sizes as follows:

1. 4000 products
2. 320 products
3. 320 products
4. 645 products
5. 645 products
6. 425 products
7. 425 products
8. 130 products
9. 130 products
10. 70 products

Now it becomes clear that between the third and fourth stages a buffer of 645 products is needed, and then it turns out that the same buffer is actually needed between the second and third stages of production. As a result, the optimal sizes of production batches by stages will be the following sequence:

1. 4000 products
2. 645 products
3. 645 products
4. 645 products
5. 645 products
6. 425 products
7. 425 products
8. 130 products
9. 130 products
10. 70 products

Those. in a steady state, any set of batches at production stages tends to such a set when at the next stage the batch size is equal to or less than the batch size of the previous stage.

Let's call this the paradox of “home canning”: first we collect all the harvest we can and put it in jars, then, on holidays, we take a jar of cucumbers from the reserves, open it, and for several days we hastily finish the open jar of cucumbers so that they have not spoiled - at each stage of “consumption” of the cucumber harvest, the batch size becomes smaller and smaller until it reaches the size of the batches in which the consumer picks up the products.

If we initially had batch sizes as follows:

1. 34 pieces
2. 540 pieces
3. 200 pieces
4. 2000 pieces

then it is quite reasonable to expect that after some time a set of batch sizes would arrive at the option

1. 2000 pieces
2. 2000 pieces
3. 2000 pieces
4. 2000 pieces

since there is no need to reconfigure the equipment of the third stage of production 10 times in order to launch one batch of 2000 identical products at the fourth stage.

Warning about conditions that remain “behind the text”

All these layouts are given for one type of product without taking into account other types of products - we simply mean that the changeover is carried out for the manufacture of a “different” type of product.

The paradox of “home canning” in its pure form can only be seen in production where there is enough production and warehouse space to store all these growing stocks. Otherwise, they will be limited by the physical scale of production, but the essence of the paradox will be the same: the size of batches at previous stages will increase until the limit of the space occupied by stocks is reached, or until this same batch size reaches the size of subsequent batches stages.

An important conclusion about the maximum optimal batch size

The size of batches at each stage of production will be no less than the size of batches of the last stage of production or the last stage of transportation of products to the customer.

Those. If you're shipping dental bicycle pumps to a customer in forty-foot containers, it makes no sense to produce them in batches of 10 rather than 50 or 1000—you'll end up needing a full container of pumps anyway.

Calculation of the minimum allowable lot size

In the logic of lean manufacturing, one of the goals of production planning is to reduce the batch size until reaching an ideal state, which is described by the concept of “one piece flow” - One Piece Flow.

If the calculation of the economically optimal batch size is done within the framework of generally accepted management logic, when certain inventory sizes are good and not evil, then in lean manufacturing, when any inventory is considered harmful to one degree or another, the question of the optimal batch size is posed a little differently: how small Can there be batches of production provided that the required level of production productivity is maintained?

Here's the calculation.

Suppose we need to produce a certain number n of products or semi-finished products in time T. The average cycle time is CT. In this case, the time that we can spend on changeovers will be equal to

Tcho = (T - n x CT)

If one changeover takes approximately the time value ChT, then we can afford a certain number of changeovers during this period of time:

Ncho = (T - n x CT) / ChT

And then the average number of products in the batch will be equal to:

Batch = n / Ncho = n x ChT / (T - n x CT)

For a maximum of changeovers performed over a certain period of time, this will be the minimum of products per batch, at which production still has time to fulfill its plan.

Here's an example.

Shift duration = 8 hours or 480 minutes

Cycle time = 1 minute/piece

Planned production of 400 products

Changeover duration 5 minutes

Batch = 450 x 5 / (480 - 400 x 1) = 450 x 5 / 80 = 29 items (round up)

For reliability, it is worth introducing an equipment availability factor to take into account the time for maintenance and repairs.

Then the formula will look like this:

Batch = n x ChT / (T x k – n x CT)

in this case, if we add an availability factor of 90% to our example, then the batch size will be equal to:

Batch = 450 x 5 / (480 x 0.9 - 400 x 1) = 450 x 5 / (432 - 400) = 450 x 5 / 32 = 71 items.

Here are some consequences from this formula:

• The larger the planned output, the fewer changeovers can be made and the larger batch sizes need to be used.
• The lower the availability factor, the fewer changeovers and the larger the batch size.
• The longer the changeover time, the fewer changeovers and the larger the batch size
• The shorter the changeover time, the more changeovers can be made and the smaller batch sizes can be used.

This formula makes two simplifications taking into account the following assumptions.